A common assumption is that a neutral process is inherently fair. However, due to natural population clustering (e.g., Democrats in cities), a randomly drawn map can still heavily favor one party. Achieving fairness may require intentional design to counteract geographic disadvantages, not just the absence of malicious intent.

While socially problematic, residential clustering of minority groups is politically advantageous. Uniformly distributed minorities risk getting 0% of seats even with significant voter share, as they can't form a majority in any single district. Clustering allows them to secure representation by creating districts they can win.

Seemingly irrational political decisions can be understood by applying a simple filter: politicians will say or do whatever they believe is necessary to get reelected. This framework decodes behavior better than assuming action is based on principle or for the public good.

A common focus in redistricting reform is preventing 'crazy-looking' districts. However, this is a red herring. A legislature can easily create visually compact, 'nice-looking' districts that are just as politically skewed, making district shape an unreliable metric for fairness.

To analyze a proposed map's fairness, mathematicians compare it to a representative sample of alternatives. They use a Markov chain—a 'random walk' making sequential changes to a map—to explore the astronomically large space of possibilities without enumerating it, creating a baseline for what 'typical' maps look like.

The combinatorial complexity of drawing district maps is vastly underestimated, even by Supreme Court justices. The number of possibilities isn't in the thousands but is astronomically large (like a googol), making it impossible to check every option and thus requiring sophisticated mathematical sampling techniques.

With over 90% of congressional districts being non-competitive, the primary election is often the only one that matters. Buttigieg argues this incentivizes candidates to appeal only to their party's extreme flank, with no need to build broader consensus for a general election.

Representation by sampling, the method used for juries, is one of two fundamental forms of democratic representation, the other being elections. While we have doubled down on elections, sampling offers a powerful, underutilized model for governance in areas like redistricting, where ordinary citizens can make fairer decisions than conflicted politicians.

Instead of single-winner districts, a powerful reform is creating larger, multi-member districts that elect several representatives (e.g., 4 districts electing 3 members each). This allows for more proportional outcomes that reflect an area's political diversity, as a minority group can win one of the multiple seats.